Consider steady, $2$D$,$ constant$-$density flow resulting from a spiraling line vortex$/$sink centered on the $z-$axis. Streamlines and velocity components are shown in the side figure. The velocity field components in polar coordinates are given by$,$ where $C$ and $K$ are constants:

$u_r=\frac{C}{r},$ and $,u_{}=\frac{K}{r}$

a$)$ Show that the velocity field satisfies the continuity equation.

b$)$ Knowing that pressure field is only a function of $r,$ use Navier$-$Stokes equations to determine $P(r).$ Take $P=P_{}$ at $r.$ Gravity acts in $z-$direction.

c$)$ Show that the flow is irrotational. The vorticity vector in polar coordinates is given by:

vec$()=\frac{1}{r}(\frac{\text{d}\text{e}\text{l}(r{u}_{})}{\text{d}\text{e}\text{l}\text{r}}-\frac{del{u}_r}{\text{d}\text{e}\text{l}})vec(k)$

d$)$ Determine the flow stream function and velocity potential function. Set the integration constants to zero.

e$)$ Show that the result of question $($b$)$ could be obtained without using Navier$-$Stokes equations. Hint: Use the fact that the flow is potential and note that $V=0$ at $r.$